My question concerns a statement on page 12 of the following paper of Baum, Connes, and Higson:

http://www.mmas.univ-metz.fr/~gnc/bibliographie/BaumConnes/Baum-Connes-Higson.pdf

about the definition of the $G$-index map from the $G$-equivariant $K$-homology group $K_0^G(X)$ to $K_0(C_r^*(G))$, where $X$ is a proper $G$-compact $G$-manifold.

Suppose $(H_+,H_-,F)$ is a $G$-equivariant abstract elliptic operator on $X$ (defined on the previous page of the same paper), so that $H_+$ and $H_-$ are Hilbert spaces equipped with unitary $G$-representations and $G$-covariant representations $\pi_{\pm}$ of $C_0(X)$, and $F$ is a certain bounded $G$-equivariant Hilbert space operator $H_+\rightarrow H_-$ satisfying a list of conditions (page 11).

One can complete of the subspace $\pi_{\pm}(C_c(X))H_\pm\subseteq H_\pm$ into a $C_r^*(G)$-module $\mathcal{H}$, as well as complete the operator $F$ into an adjointable operator $\mathcal{F}$ on $\mathcal{H}$. It is then stated that

*"thanks to the axioms for an abstract elliptic operator (and the fact that $X$ is $G$-compact)",*

$\mathcal{F}$ is in fact a Fredholm operator. I'm looking for some clarification and confirmation on two questions (the first is more of a sanity check):

When $H = L^2(X;E)$, where $E$ is a vector bundle over $X$, is the completion $\mathcal{H}$ the same as the the completion of $C_c(X,E)$ (viewed as a $C_c(G)$-module) into a $C_r^*(G)$-module?

How does one prove Fredholmness of $\mathcal{F}$ using the axioms for an abstract elliptic operator and $G$-compactness of $X$, as stated in the paper?

I have seen Kasparov (in his recent JNCG paper) show Fredholmness of classes in $K_0^G(X)$ represented by *elliptic* operators, but Baum-Connes-Higson seem to assert that this is true for any $K$-homology cycle.

Any references/proofs would be great. Thanks!